Optimal. Leaf size=157 \[ \frac {b^2}{16 c^2 d^3 (1+c x)^2}-\frac {5 b^2}{16 c^2 d^3 (1+c x)}+\frac {5 b^2 \tanh ^{-1}(c x)}{16 c^2 d^3}+\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^2 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2} \]
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Rubi [A]
time = 0.16, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {37, 6085,
6063, 641, 46, 213, 6095} \begin {gather*} -\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (c x+1)}+\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (c x+1)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^2 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (c x+1)^2}-\frac {5 b^2}{16 c^2 d^3 (c x+1)}+\frac {b^2}{16 c^2 d^3 (c x+1)^2}+\frac {5 b^2 \tanh ^{-1}(c x)}{16 c^2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 46
Rule 213
Rule 641
Rule 6063
Rule 6085
Rule 6095
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{(d+c d x)^3} \, dx &=\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}-(2 b c) \int \left (\frac {a+b \tanh ^{-1}(c x)}{4 c^2 d^3 (1+c x)^3}-\frac {3 \left (a+b \tanh ^{-1}(c x)\right )}{8 c^2 d^3 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{8 c^2 d^3 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}+\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{4 c d^3}-\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx}{2 c d^3}+\frac {(3 b) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{4 c d^3}\\ &=\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^2 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}-\frac {b^2 \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx}{4 c d^3}+\frac {\left (3 b^2\right ) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{4 c d^3}\\ &=\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^2 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}-\frac {b^2 \int \frac {1}{(1-c x) (1+c x)^3} \, dx}{4 c d^3}+\frac {\left (3 b^2\right ) \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{4 c d^3}\\ &=\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^2 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}-\frac {b^2 \int \left (\frac {1}{2 (1+c x)^3}+\frac {1}{4 (1+c x)^2}-\frac {1}{4 \left (-1+c^2 x^2\right )}\right ) \, dx}{4 c d^3}+\frac {\left (3 b^2\right ) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{4 c d^3}\\ &=\frac {b^2}{16 c^2 d^3 (1+c x)^2}-\frac {5 b^2}{16 c^2 d^3 (1+c x)}+\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^2 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}+\frac {b^2 \int \frac {1}{-1+c^2 x^2} \, dx}{16 c d^3}-\frac {\left (3 b^2\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{8 c d^3}\\ &=\frac {b^2}{16 c^2 d^3 (1+c x)^2}-\frac {5 b^2}{16 c^2 d^3 (1+c x)}+\frac {5 b^2 \tanh ^{-1}(c x)}{16 c^2 d^3}+\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (1+c x)^2}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^2 d^3 (1+c x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^2 d^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 150, normalized size = 0.96 \begin {gather*} \frac {2 \left (8 a^2+4 a b+b^2\right )-2 \left (16 a^2+12 a b+5 b^2\right ) (1+c x)-8 b (b (2+3 c x)+a (4+8 c x)) \tanh ^{-1}(c x)+4 b^2 \left (-1-2 c x+3 c^2 x^2\right ) \tanh ^{-1}(c x)^2-b (12 a+5 b) (1+c x)^2 \log (1-c x)+b (12 a+5 b) (1+c x)^2 \log (1+c x)}{32 c^2 d^3 (1+c x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(390\) vs.
\(2(143)=286\).
time = 0.36, size = 391, normalized size = 2.49
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (-\frac {1}{c x +1}+\frac {1}{2 \left (c x +1\right )^{2}}\right )}{d^{3}}-\frac {b^{2} \arctanh \left (c x \right )^{2}}{d^{3} \left (c x +1\right )}+\frac {b^{2} \arctanh \left (c x \right )^{2}}{2 d^{3} \left (c x +1\right )^{2}}+\frac {b^{2} \arctanh \left (c x \right )}{4 d^{3} \left (c x +1\right )^{2}}-\frac {3 b^{2} \arctanh \left (c x \right )}{4 d^{3} \left (c x +1\right )}+\frac {3 b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{8 d^{3}}-\frac {3 b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{8 d^{3}}+\frac {3 b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16 d^{3}}-\frac {3 b^{2} \ln \left (c x -1\right )^{2}}{32 d^{3}}-\frac {3 b^{2} \ln \left (c x +1\right )^{2}}{32 d^{3}}-\frac {3 b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16 d^{3}}+\frac {3 b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{16 d^{3}}+\frac {b^{2}}{16 d^{3} \left (c x +1\right )^{2}}-\frac {5 b^{2}}{16 d^{3} \left (c x +1\right )}+\frac {5 b^{2} \ln \left (c x +1\right )}{32 d^{3}}-\frac {5 b^{2} \ln \left (c x -1\right )}{32 d^{3}}-\frac {2 a b \arctanh \left (c x \right )}{d^{3} \left (c x +1\right )}+\frac {a b \arctanh \left (c x \right )}{d^{3} \left (c x +1\right )^{2}}+\frac {a b}{4 d^{3} \left (c x +1\right )^{2}}-\frac {3 a b}{4 d^{3} \left (c x +1\right )}+\frac {3 a b \ln \left (c x +1\right )}{8 d^{3}}-\frac {3 a b \ln \left (c x -1\right )}{8 d^{3}}}{c^{2}}\) | \(391\) |
default | \(\frac {\frac {a^{2} \left (-\frac {1}{c x +1}+\frac {1}{2 \left (c x +1\right )^{2}}\right )}{d^{3}}-\frac {b^{2} \arctanh \left (c x \right )^{2}}{d^{3} \left (c x +1\right )}+\frac {b^{2} \arctanh \left (c x \right )^{2}}{2 d^{3} \left (c x +1\right )^{2}}+\frac {b^{2} \arctanh \left (c x \right )}{4 d^{3} \left (c x +1\right )^{2}}-\frac {3 b^{2} \arctanh \left (c x \right )}{4 d^{3} \left (c x +1\right )}+\frac {3 b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{8 d^{3}}-\frac {3 b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{8 d^{3}}+\frac {3 b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16 d^{3}}-\frac {3 b^{2} \ln \left (c x -1\right )^{2}}{32 d^{3}}-\frac {3 b^{2} \ln \left (c x +1\right )^{2}}{32 d^{3}}-\frac {3 b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16 d^{3}}+\frac {3 b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{16 d^{3}}+\frac {b^{2}}{16 d^{3} \left (c x +1\right )^{2}}-\frac {5 b^{2}}{16 d^{3} \left (c x +1\right )}+\frac {5 b^{2} \ln \left (c x +1\right )}{32 d^{3}}-\frac {5 b^{2} \ln \left (c x -1\right )}{32 d^{3}}-\frac {2 a b \arctanh \left (c x \right )}{d^{3} \left (c x +1\right )}+\frac {a b \arctanh \left (c x \right )}{d^{3} \left (c x +1\right )^{2}}+\frac {a b}{4 d^{3} \left (c x +1\right )^{2}}-\frac {3 a b}{4 d^{3} \left (c x +1\right )}+\frac {3 a b \ln \left (c x +1\right )}{8 d^{3}}-\frac {3 a b \ln \left (c x -1\right )}{8 d^{3}}}{c^{2}}\) | \(391\) |
risch | \(\frac {b^{2} \left (3 c^{2} x^{2}-2 c x -1\right ) \ln \left (c x +1\right )^{2}}{32 c^{2} d^{3} \left (c x +1\right )^{2}}-\frac {b \left (3 b \,x^{2} \ln \left (-c x +1\right ) c^{2}-2 b c x \ln \left (-c x +1\right )+16 c x a +6 b c x -b \ln \left (-c x +1\right )+8 a +4 b \right ) \ln \left (c x +1\right )}{16 c^{2} d^{3} \left (c x +1\right )^{2}}+\frac {-16 a^{2}-16 a b -8 b^{2}+12 \ln \left (-c x -1\right ) a b -12 a b \ln \left (c x -1\right )-b^{2} \ln \left (-c x +1\right )^{2}+8 b^{2} \ln \left (-c x +1\right )+3 b^{2} c^{2} x^{2} \ln \left (-c x +1\right )^{2}-5 b^{2} \ln \left (c x -1\right )-24 a b c x +16 b \ln \left (-c x +1\right ) a +32 a b c x \ln \left (-c x +1\right )-12 \ln \left (c x -1\right ) a b \,c^{2} x^{2}+5 b^{2} c^{2} \ln \left (-c x -1\right ) x^{2}+5 b^{2} \ln \left (-c x -1\right )+12 b \,c^{2} \ln \left (-c x -1\right ) x^{2} a +10 \ln \left (-c x -1\right ) b^{2} c x -10 \ln \left (c x -1\right ) b^{2} c x +12 b^{2} c x \ln \left (-c x +1\right )-2 b^{2} c x \ln \left (-c x +1\right )^{2}-10 b^{2} c x -32 a^{2} c x +24 \ln \left (-c x -1\right ) a b c x -24 \ln \left (c x -1\right ) a b c x -5 \ln \left (c x -1\right ) b^{2} c^{2} x^{2}}{32 c^{2} d^{3} \left (c x +1\right )^{2}}\) | \(433\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 429 vs.
\(2 (143) = 286\).
time = 0.28, size = 429, normalized size = 2.73 \begin {gather*} -\frac {{\left (2 \, c x + 1\right )} b^{2} \operatorname {artanh}\left (c x\right )^{2}}{2 \, {\left (c^{4} d^{3} x^{2} + 2 \, c^{3} d^{3} x + c^{2} d^{3}\right )}} - \frac {1}{8} \, {\left (c {\left (\frac {2 \, {\left (3 \, c x + 2\right )}}{c^{5} d^{3} x^{2} + 2 \, c^{4} d^{3} x + c^{3} d^{3}} - \frac {3 \, \log \left (c x + 1\right )}{c^{3} d^{3}} + \frac {3 \, \log \left (c x - 1\right )}{c^{3} d^{3}}\right )} + \frac {8 \, {\left (2 \, c x + 1\right )} \operatorname {artanh}\left (c x\right )}{c^{4} d^{3} x^{2} + 2 \, c^{3} d^{3} x + c^{2} d^{3}}\right )} a b - \frac {1}{32} \, {\left (4 \, c {\left (\frac {2 \, {\left (3 \, c x + 2\right )}}{c^{5} d^{3} x^{2} + 2 \, c^{4} d^{3} x + c^{3} d^{3}} - \frac {3 \, \log \left (c x + 1\right )}{c^{3} d^{3}} + \frac {3 \, \log \left (c x - 1\right )}{c^{3} d^{3}}\right )} \operatorname {artanh}\left (c x\right ) + \frac {{\left (3 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x + 1\right )^{2} + 3 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 10 \, c x - {\left (5 \, c^{2} x^{2} + 10 \, c x + 6 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 5\right )} \log \left (c x + 1\right ) + 5 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 8\right )} c^{2}}{c^{6} d^{3} x^{2} + 2 \, c^{5} d^{3} x + c^{4} d^{3}}\right )} b^{2} - \frac {{\left (2 \, c x + 1\right )} a^{2}}{2 \, {\left (c^{4} d^{3} x^{2} + 2 \, c^{3} d^{3} x + c^{2} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 164, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (16 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} c x - {\left (3 \, b^{2} c^{2} x^{2} - 2 \, b^{2} c x - b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} + 16 \, a^{2} + 16 \, a b + 8 \, b^{2} - {\left ({\left (12 \, a b + 5 \, b^{2}\right )} c^{2} x^{2} - 2 \, {\left (4 \, a b + b^{2}\right )} c x - 4 \, a b - 3 \, b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{32 \, {\left (c^{4} d^{3} x^{2} + 2 \, c^{3} d^{3} x + c^{2} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2} x}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {b^{2} x \operatorname {atanh}^{2}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {2 a b x \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 226, normalized size = 1.44 \begin {gather*} \frac {1}{64} \, c {\left (\frac {2 \, {\left (\frac {2 \, {\left (c x + 1\right )} b^{2}}{c x - 1} + b^{2}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )}^{2} c^{3} d^{3}} + \frac {2 \, {\left (\frac {8 \, {\left (c x + 1\right )} a b}{c x - 1} + 4 \, a b + \frac {4 \, {\left (c x + 1\right )} b^{2}}{c x - 1} + b^{2}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{2} c^{3} d^{3}} + \frac {{\left (\frac {16 \, {\left (c x + 1\right )} a^{2}}{c x - 1} + 8 \, a^{2} + \frac {16 \, {\left (c x + 1\right )} a b}{c x - 1} + 4 \, a b + \frac {8 \, {\left (c x + 1\right )} b^{2}}{c x - 1} + b^{2}\right )} {\left (c x - 1\right )}^{2}}{{\left (c x + 1\right )}^{2} c^{3} d^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.69, size = 405, normalized size = 2.58 \begin {gather*} -\frac {16\,a\,b+17\,b^2\,\ln \left (c\,x+1\right )-17\,b^2\,\ln \left (1-c\,x\right )+b^2\,{\ln \left (c\,x+1\right )}^2+b^2\,{\ln \left (1-c\,x\right )}^2-28\,b^2\,\mathrm {atanh}\left (c\,x\right )+16\,a^2+8\,b^2+16\,a\,b\,\ln \left (c\,x+1\right )-16\,a\,b\,\ln \left (1-c\,x\right )-2\,b^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-24\,a\,b\,\mathrm {atanh}\left (c\,x\right )+32\,a^2\,c\,x+10\,b^2\,c\,x+30\,b^2\,c\,x\,\ln \left (c\,x+1\right )-30\,b^2\,c\,x\,\ln \left (1-c\,x\right )-3\,b^2\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2-3\,b^2\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^2-28\,b^2\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+2\,b^2\,c\,x\,{\ln \left (c\,x+1\right )}^2+2\,b^2\,c\,x\,{\ln \left (1-c\,x\right )}^2-56\,b^2\,c\,x\,\mathrm {atanh}\left (c\,x\right )+9\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )-9\,b^2\,c^2\,x^2\,\ln \left (1-c\,x\right )+24\,a\,b\,c\,x+32\,a\,b\,c\,x\,\ln \left (c\,x+1\right )-32\,a\,b\,c\,x\,\ln \left (1-c\,x\right )-4\,b^2\,c\,x\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )-24\,a\,b\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )-48\,a\,b\,c\,x\,\mathrm {atanh}\left (c\,x\right )+6\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{32\,c^2\,d^3\,{\left (c\,x+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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